It can be the battery in the switch, in this case have the guy in California (GM5 repair) who is very experienced at replacement, sort his for you. I can recommend him.

The other thing is the FZV antenna module (receives and process the key signal). In my case it was the FZV module. I found a good, used module and after replacement - my remote keys work. Make sure you check your FZV module, as there are different frequencies used depending on where you are in the world.

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2002 Euro X5 (E53) 3.0 diesel (M57)
5 speed manual with UUC short shifter.
Pre-xDrive NV125 transfer case (chain & gears).
Factory Xenon headlights, Spyder LED tail lights
Updated vortex crankcase breather.
Motorsport dead pedal, factory fire extinguisher, factory first aid kit, factory F&R mud flaps, factory PDC

Bilstein B6 shocks F&R
Redline oil has replaced "lifetime" fluids (F&R diff, manual transmission, transfer case and P/S).
Dimple magnetic drain plugs in all compartments.